## Elements of Differentiable Dynamics and Bifurcation Theory by David Ruelle

By David Ruelle

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23 If (x",x') G E^(R) x E'0(R), R small, the distance of (x",χ') to (χ",φ"{χ")) is multiplied by a factor < 1 fk(x"\x') when we apply / . 1 V" = f)n^0f E0(R). 24 While Hadamard's method applies most naturally to the unstable manifold, Perron proved the smoothness of the stable manifold by starting from the definition {x : fnx —> 0}. The stable and unstable manifold theorem can also be proven by a clever application of the implicit function theorem 25 or by other techniques. In the analytic finite dimensional case, 1-dimensional unstable manifolds can be obtained in 21 £ i is the set of functions E"(R) ·->· Ef0(R) vanishing at 0 and with Lipschitz constant ^ 1, which means that \\ψ(χ) — Ψ(ν)\\ ^ \\x — y\\ for all x, y.

A model of magnetic hysteresis can be given in terms of bifurcations of a dynamical system (see Erber, Guralnik, and Latal [1]). 3. 41 Incidentally, let us remark that one often says that the property " P " is nongeneric to mean that "non P " is generic (this is definitely an abuse of language). Also, the word generic is sometimes used in a loose sense rather than with the precise meaning defined above. 8. Generic Properties of Diffeomorphisms and Flows on a Compact Manifold Below we give two famous examples, one due to Kupka-Smale and one to Pugh, of generic properties of diffeomorphisms and flows on a compact manifold M.

We define the global stable and unstable manifolds by W~ = \xeM: lim fnx = a] , Wa = \xo> 3(sn)n>o with / x n + i = #n and lim x n = a \ . (a) Show that If / is a diffeomorphism, each f±nV± is C r . If Tf : T M ·-+ TM is surjective and V~ has finite codimension (or M is a Hubert 49 Differentiable Dynamical Systems manifold), then f~nV~ is CT. If Tf : TM \-+ TM is injective and V+ finite dimensional, then / n V + is CT. The sets W^ are thus locally smooth manifolds. ] Note: If / is a diffeomorphism of a finite dimensional manifold, one can show that W^ is the image of V^ by an injective CT immersion tangent to the identity at a.